#7005650

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,38,44)(19,29,34,41,33,24,32,42)(20,26,36,43,21,37,35,25)(30,46,47,39,31,40,48,45);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,31)(22,34)(23,33)(25,39)(26,40)(27,42)(28,29)(32,38)(35,48)(36,47)(37,46)(41,44)(43,45);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,25,28,43)(19,30,33,31)(22,35,38,36)(24,39,29,45)(26,27,37,44)(32,47,34,48)(40,42,46,41);(3,8)(6,7)(11,16)(13,15)(17,21,23,20)(18,26,28,37)(19,31,33,30)(22,35,38,36)(24,40,29,46)(25,27,43,44)(32,47,34,48)(39,42,45,41);(10,14)(13,15)(17,22,23,38)(18,27,28,44)(19,32,33,34)(20,35,21,36)(24,41,29,42)(25,37,43,26)(30,47,31,48)(39,40,45,46)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6912846);